The asymptotic distributions of the statistics we propose are based on extreme and intermediate asymptotic approximations. Such approximations are suited to inference on extreme quantiles, and complement the central approximations constructed by [1]} for conducting inference on central quantiles of UIH.[1]} also study the problem of estimating the distribution of UIH given only noisy estimates; this is the forward problem, compared to the inverse problem of estimating quantiles. The problem of estimating the distribution of UIH in the presence of estimation noise is also tightly linked to the normal means problem (see [3]}, [4]} for some recent contributions).
The three asymptotic approximations differ in how the quantile of interest is modeled as a function of the cross-sectional size.
Extreme order approximations model the quantile of interest as drifting to 0 or 1 at rate proportional to the cross-sectional dimension.This approach is related to the fixed-\(k\)  tail inference methods of [5]}, which they develop in a noiseless setting. The resulting
confidence intervals and tests for extreme quantiles of UIH are based on a ratio of extreme order statistics of the estimated UIH.
The distribution of such ratios depends on the extreme value (EV) index of the distribution of latent true UIH, and thus is non-pivotal. The corresponding critical values can be obtained by subsampling or by simulation. For simulating the critical values, we propose using the [6]} estimator for the EV index and prove its consistency. We establish consistency of the resulting critical values for both methods. In intermediate order approximations, the quantile of interest drifts to 0 or 1 at a rate slower than the cross-sectional sample size. In this case we base confidence intervals on a ratio statistic of intermediate order statistics. This ratio exploits a certain knife-edge bias property between two perfectly dependent random variables to normalize the statistic.This result may also be of interest in the noiseless case.
In contrast, central order approximations are based on the central limit theorem for quantiles; the quantile of interest is modeled as fixed.
